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In June 2021, the Pacific Northwest experienced a heatwave that broke all previous records. Estimated return levels based on observations up to the year before the event suggested that reaching such high temperatures is not possible in today's climate. We here assess the suitability of the prevalent statistical approach by analyzing extreme temperature events in climate model large ensemble and synthetic extreme value data. We demonstrate that the method is subject to biases, as high return levels are generally underestimated and, correspondingly, the return period of low‐likelihood heatwave events is overestimated, if the underlying extreme value distribution is derived from a short historical record. These biases have even increased in recent decades due to the emergence of a pronounced climate change signal. Furthermore, if the analysis is triggered by an extreme event, the implicit selection bias affects the likelihood assessment depending on whether the event is included in the modeling. Plain Language Summary In June 2021, the Pacific Northwest experienced a record‐breaking heatwave event. Based on historical data, the scientific community has applied statistical models to understand how likely this event was to occur. However, due to the record‐shattering nature of this particular heatwave, the model suggested that reaching such high temperatures should not have been possible. In this study, we evaluate the accuracy of these statistical models in describing the occurrence probability of extreme events. We find that the current models tend to underestimate the occurrence probability and that the bias has become more pronounced in recent years due to climate change. Finally, we assess how the way extreme events are included in the model can also affect the accuracy of estimates. Key Points Standard return period estimates of temperature extremes are systematically overestimated in short records under non‐stationary conditions The small‐sample bias in maximum likelihood estimates is found both for extremes in climate model data and in synthetic data experiments Future analysis should account for the statistical implications of the selection bias if the analysis is triggered by an extreme event

In our thesis, we employ a forth and means of traveling flats for an equation set including completely non-linear non-local operators in B 1 (0). Under the conditions ζ η ∈ C l o c 1 , 1 ∩ L α and υ η ∈ C l o c 1 , 1 ∩ L β , the symmetric character of radial direction and monotone character of plus solutions for an equation set are proved. In order to get this result, we use the extreme value theory of narrow area for the equation set and pivotal ingredients for carrying on the way of traveling flats.

Coastal regions are dynamic areas that often lie at the junction of different natural hazards. Extreme events such as storm surges and high precipitation are significant sources of concern for flood management. As climatic changes and sea-level rise put further pressure on these vulnerable systems, there is a need for a better understanding of the implications of compounding hazards. Recent computational advances in hydraulic modelling offer new opportunities to support decision-making and adaptation. Our research makes use of recently released features in the HEC-RAS version 5.0 software to develop an integrated 1D–2D hydrodynamic model. Using extreme value analysis with the Peaks-Over-Threshold method to define extreme scenarios, the model was applied to the eastern coast of the UK. The sensitivity of the protected wetland known as the Broads to a combination of fluvial, tidal and coastal sources of flooding was assessed, accounting for different rates of twenty-first century sea-level rise up to the year 2100. The 1D–2D approach led to a more detailed representation of inundation in coastal urban areas, while allowing for interactions with more fluvially dominated inland areas to be captured. While flooding was primarily driven by increased sea levels, combined events exacerbated flooded area by 5–40% and average depth by 10–32%, affecting different locations depending on the scenario. The results emphasise the importance of catchment-scale strategies that account for potentially interacting sources of flooding.

Quantile mapping bias correction algorithms are commonly used to correct systematic distributional biases in precipitation outputs from climate models. Although they are effective at removing historical biases relative to observations, it has been found that quantile mapping can artificially corrupt future model-projected trends. Previous studies on the modification of precipitation trends by quantile mapping have focused on mean quantities, with less attention paid to extremes. This article investigates the extent to which quantile mapping algorithms modify global climate model (GCM) trends in mean precipitation and precipitation extremes indices. First, a bias correction algorithm, quantile delta mapping (QDM), that explicitly preserves relative changes in precipitation quantiles is presented. QDM is compared on synthetic data with detrended quantile mapping (DQM), which is designed to preserve trends in the mean, and with standard quantile mapping (QM). Next, methods are applied to phase 5 of the Coupled Model Intercomparison Project (CMIP5) daily precipitation projections over Canada. Performance is assessed based on precipitation extremes indices and results from a generalized extreme value analysis applied to annual precipitation maxima. QM can inflate the magnitude of relative trends in precipitation extremes with respect to the raw GCM, often substantially, as compared to DQM and especially QDM. The degree of corruption in the GCM trends by QM is particularly large for changes in long period return values. By the 2080s, relative changes in excess of +500% with respect to historical conditions are noted at some locations for 20-yr return values, with maximum changes by DQM and QDM nearing +240% and +140%, respectively, whereas raw GCM changes are never projected to exceed +120%.

In July 2021 extreme rainfall across Western Europe caused severe flooding and substantial impacts, including over 200 fatalities and extensive infrastructure damage within Germany and the Benelux countries. After the event, a hydrological assessment and a probabilistic event attribution analysis of rainfall data were initiated and complemented by discussing the vulnerability and exposure context. The global mean surface temperature (GMST) served as a covariate in a generalised extreme value distribution fitted to observational and model data, exploiting the dependence on GMST to estimate how anthropogenic climate change affects the likelihood and severity of extreme events. Rainfall accumulations in Ahr/Erft and the Belgian Meuse catchment vastly exceeded previous observed records. In regions of that limited size the robust estimation of return values and the detection and attribution of rainfall trends are challenging. However, for the larger Western European region it was found that, under current climate conditions, on average one rainfall event of this magnitude can be expected every 400 years at any given location. Consequently, within the entire region, events of similar magnitude are expected to occur more frequently than once in 400 years. Anthropogenic climate change has already increased the intensity of the maximum 1-day rainfall event in the summer season by 3–19 %. The likelihood of such an event to occur today compared to a 1.2 ∘C cooler climate has increased by a factor of 1.2–9. Models indicate that intensity and frequency of such events will further increase with future global warming. While attribution of small-scale events remains challenging, this study shows that there is a robust increase in the likelihood and severity of rainfall events such as the ones causing extreme impacts in July 2021 when considering a larger region.

Central ideas from thermal biology, including thermal performance curves and tolerances, have been widely used to evaluate how changes in environmental means and variances generate changes in fitness, selection and microevolution in response to climate change. We summarize the opportunities and challenges for extending this approach to understanding the consequences of extreme climatic events. Using statistical tools from extreme value theory, we show how distributions of thermal extremes vary with latitude, time scale and climate change. Second, we review how performance curves and tolerances have been used to predict the fitness and evolutionary responses to climate change and climate gradients. Performance curves and tolerances change with prior thermal history and with time scale, complicating their use for predicting responses to thermal extremes. Third, we describe several recent case studies showing how infrequent extreme events can have outsized effects on the evolution of performance curves and heat tolerance. A key issue is whether thermal extremes affect reproduction or survival, and how these combine to determine overall fitness. We argue that a greater focus on tails—in the distribution of environmental extremes, and in the upper ends of performance curves—is needed to understand the consequences of extreme events. This article is part of the themed issue ‘Behavioural, ecological and evolutionary responses to extreme climatic events’.

Extreme climate events (ECEs) like heavy rainfall and heatwaves significantly impact society, and climate change is altering their magnitude and frequency. Generalized Extreme Value (GEV) distributions help quantify these ECEs and guide human system design. We train a machine learning (ML) model using a set of arbitrary GEV distributions to estimate the sample size required to determine a return value with specific uncertainty. For ECEs like heatwaves with a negative GEV shape parameter the maximum extreme temperatures of heatwaves are bounded and fewer samples are needed to estimate the return value to given uncertainty than rainfall extremes which have positive shape parameter with unbounded extreme values. For example, if a 1‐in‐20‐year heatwave event requires 400 samples to estimate return value to ±$\\pm $ 1% uncertainty, one would need 20 different 20‐year simulations. Achieving such quantities will require extensive climate downscaling simulations, potentially aided by ML‐based downscaling methods to increase the ensemble size. Plain Language Summary Generalized Extreme Value (GEV) distribution is a common way to characterize extreme climate events from climate data sets. We develop a machine learning model to estimate the sample size required to determine a return value to a prescribed uncertainty for an arbitrary set of GEV parameters. For the expected GEV parameters relevant to climate variables, the number of years needed to quantify the annual return value of low probability events (e.g., a 1 in 100‐year event) can easily exceed 1000s of years of simulations to get sufficiently accurate estimates of the return value to differentiate it from a more likely event (e.g., 1 in 50‐year event). By knowing the sample size, one can start to design climate change simulation experiments with sufficient simulated years to detect how annual return values are changing with climate change. Key Points Generalized extreme value (GEV) distributions are a common way to characterize extreme climate events in climate data sets We develop a machine learning model to quantify the sample size needed to estimate return value of GEV distribution to specific uncertainty Knowing how return values are influenced by sample size will help to design experiments to attribute climate change impacts on extreme events

The results of probabilistic event attribution studies depend on the choice of the extreme value statistics used in the analysis, particularly with the arbitrariness in the selection of appropriate thresholds to define extremes. We bypass this issue by using the Extended Generalized Pareto Distribution (ExtGPD), which jointly models low precipitation with a generalized Pareto distribution and extremes with a different Pareto tail, to conduct daily precipitation attribution across the contiguous United States (CONUS). We apply the ExtGPD to 12 general circulation models from the Coupled Model Intercomparison Project Phase 6 and compare counterfactual scenarios with and without anthropogenic emissions. Observed precipitation by the Climate Prediction Center is used for evaluating the GCMs. We find that greenhouse gases rather than natural variability can explain the observed magnitude of extreme daily precipitation, especially in the temperate regions. Our results highlight an unambiguous linkage of anthropogenic emissions to daily precipitation extremes across CONUS. Plain Language Summary We investigate how human‐induced emissions affect daily rainfall extremes across the United States. The attribution of an extreme event to human‐induced emissions depends on the selected extreme event statistics, with setting a threshold to define what counts as an extreme event remaining a major challenge. To overcome this, we used the Extended Generalized Pareto Distribution (ExtGPD) that jointly models both low and heavy rainfall events without defining a threshold, providing a more complete picture of the full distribution including extremes. We fitted the ExtGPD to 12 general circulation models and compared scenarios with and without human‐induced emissions. Our findings suggest that human emissions are responsible for the observed intensity of daily rainfall extremes across the United States, especially in regions with temperate climates, and that these extremes would have been smaller without greenhouse gases. Key Points We apply the Extended Generalized Pareto Distribution for probabilistic event attribution to bypass issues with threshold specification Anthropogenic emissions alone could exacerbate the observed magnitude of extreme daily precipitation across the United States The study underscores the urgent need for mitigation, revealing a clear link between anthropogenic activities and extreme precipitation

Research in imbalanced domain learning has almost exclusively focused on solving classification tasks for accurate prediction of cases labelled with a rare class. Approaches for addressing such problems in regression tasks are still scarce due to two main factors. First, standard regression tasks assume each domain value as equally important. Second, standard evaluation metrics focus on assessing the performance of models on the most common values of data distributions. In this paper, we present an approach to tackle imbalanced regression tasks where the objective is to predict extreme (rare) values. We propose an approach to formalise such tasks and to optimise/evaluate predictive models, overcoming the factors mentioned and issues in related work. We present an automatic and non-parametric method to obtain relevance functions, building on the concept of relevance as the mapping of target values into non-uniform domain preferences. Then, we propose SERA, a new evaluation metric capable of assessing the effectiveness and of optimising models towards the prediction of extreme values while penalising severe model bias. An experimental study demonstrates how SERA provides valid and useful insights into the performance of models in imbalanced regression tasks.

Heat waves such as the one in Europe 2003 have severe consequences for the economy, society, and ecosystems. It is unclear whether temperatures could have exceeded these anomalies even without further climate change. Developing storylines and quantifying the highest possible temperature levels is challenging given the lack of a long homogeneous time series and methodological framework to assess them. Here, we address this challenge by analyzing summer temperatures in a nearly 5000-yr preindustrial climate model simulation, performed with the Community Earth System Model CESM1. To assess how anomalous temperatures could get, we compare storylines generated by three different methods: 1) a return-level estimate, deduced from a generalized extreme value distribution; 2) a regression model, based on dynamic and thermodynamic heat wave drivers; and 3) a novel ensemble boosting method, generating large samples of reinitialized extreme heat waves in the long climate simulation. All methods provide consistent temperature estimates, suggesting that historical exceptional heat waves such as those in Chicago in 1995, Europe in 2003, and Russia in 2010 could have been substantially exceeded even in the absence of further global warming. These estimated unseen heat waves are caused by the same drivers as moderate observed events, but with more anomalous patterns. Moreover, altered contributions of circulation and soil moisture to temperature anomalies include amplified feedbacks in the surface energy budget. The methodological framework of combining different storyline approaches of heat waves with magnitudes beyond the observational record may ultimately contribute to adaptation and to the stress testing of ecosystems or socioeconomic systems to increase resilience to extreme climate stressors.